Resolution in the Case of the Hydrogen Atom of an Improved Dirac Equation

The improved Dirac equation is completely solved in the case of the hydrogen atom. A method of separation of variables in spherical coordinates is used. The angular functions are the same as with the linear Dirac equation: they account for the spin 1/2 of the electron. The existence of a probability density governs the radial equations. This gives all the quantum numbers required by spectroscopy, the true number of energy levels and the true levels obtained by Sommerfeld’s formula.


Introduction
The improved Dirac equation was obtained from Lochak's theory of a leptonic magnetic monopole [1] [2] [3]. In this theory, the quantum wave has two U(1) gauge invariances. The mass term of the leptonic magnetic monopole is then able to replace the usual mass term of the Dirac equation. First, the non-linear mass term was read in the frame of Hestenes's space-time algebra [4] [5] [6]- [11]. Cl [12]- [34].
Since 1928, the relativistic invariance of Dirac's theory used the previous Pauli matrices for the spin of the electron: the space-time variable ( ) Journal of Modern Physics was replaced by This is equivalent to saying that the three Pauli matrices: form a orthogonal oriented basis in space. We shall put arrows on vectors in space, so any vector reads The geometric algebra of space det .
The main reason to use the geometric algebra 3 Cl is the ability to read all relativistic quantum physics in this algebra: the electron wave is a function of space and time in 3 Cl : The link between 3 Cl and the complex formalism is simple only if we use the left and right Weyl spinors η and ξ , by letting: . The usual formalism uses a ψ and µ γ matrices defined as following: Our improved wave equation of the electron, which has the Dirac equation as linear approximation, reads [6]- [19]: where q e c =  , Multiplying (11) where β is the Yvon-Takabayasi angle. In the usual formalism of complex

4
× matrices, much more complicated than the Pauli algebra, this wave equation reads:

Separating Variables
To solve the Dirac equation or the improved equation in the case of the hydrogen atom, two methods exist. We shall use here, not the initial method based on the non-relativistic approximation of the wave equation, but the new method invented by H. Krüger [35], separating the variables in spherical coordinates: We use the following notation:  . sin H. Krüger obtained the remarkable identity: which with gives also: Aiming for the separation of the temporal variable 0 x ct = and the angular variable ϕ from the radial variable r and the other angular variable θ , we let: where X is a function (with value in the Pauli algebra) of only r and θ , cE  is the energy of the electron, δ is an arbitrary phase (that plays no role here because the wave equations are electric gauge-invariant), and λ is a real constant which will be interpreted as the magnetic quantum number (we name here this quantum number λ because m is used in the mass term). We then get: We also have: we get: . sin Then with the form (26) of the wave, the Yvon-Takabayasi angle β depends neither on the time nor on the ϕ angle. It depends only on r and θ . Hence the separation of variables can be similarly obtained for the Dirac equation or for the improved equation. We have: We then get: For the hydrogen atom we have: where α is the fine structure constant. We have: Also the wave Equation (15) becomes: which means: The Dirac equation, in contrast, gives: Now we let, z being the complex conjugate of z: where , , , a b c d are functions with complex values of the real variables r and θ .
We get: We then obtain the following equations: Therefore, the improved equation is equivalent to: Conjugating the equations containing the terms of the right columns we get the system: Then if a κ constant exists such that: To get the system equivalent to the Dirac equation it is enough to suppress the β angle. This does not change the angular system (58), while in the place of (59) we get the system:

Kinetic Momentum Operators
We established in [8] We indeed have also: From (26) we have the following equivalence for 3 J : Then the φ wave satisfying (26) is a proper vector of 3 J and λ is the magnetic quantum number. Moreover for a wave φ satisfying (26), we have: if and only if: And (58) implies at the second order: ; with the definition of S in (21) and with (26) The angular system (58) is then equivalent [6] to the differential equation: The change of variable: gives then the differential equation of the Gegenbauer polynomials 1 : And we get, as only integrable function: n a a a a a n term is a factor of U and V, its argument may be absorbed by the δ of (26), and its modulus may be transferred to the radial functions. We can then let ( ) Since we have the (71) conditions on λ and κ , an integer n always exists such as: When we solve the Dirac equation with Darwin's method, that means with the ad-hoc operators, we get some Legendre polynomials and spherical harmonics. Here, working with φ , which is equivalent to working with the Weyl spinors ξ and η , we get the Gegenbauer polynomials, and it is the degree of these polynomials that gives the needed quantum number.

Resolution of the Linear Radial System
We employ the following transformations: ; .
The (60) system becomes: And the (59) system becomes: Since this radial system has the same asymptotic behavior as its linear approximation (84) we are left with the following as the only integrable solution: We now study the case where 0 n > . The (88) equation is equivalent to: We then arrive at three kinds of systems: index 0, index between 0 and n, and index n. For the null index the system is independent of β : This system is the same as in the linear case. We obtain a non-null solution only if the determinant is null, then only if s satisfies: Using (97) we get: Substituting these relations into (105) we obtain: Since 0 n a ≠ we get:

Constant Radial Polynomials
To arrive at all the results of the Dirac equation, there is one last thing to explain: . For this we must return to the particular case where the radial polynomial is reduced to a constant. We begin directly from (84), and we let:

Concluding Remarks
Sommerfeld's formula (114) for energy levels does not account for the Lamb effect which is, if 0 n > , a very small shift between the energy levels with the same quantum numbers but with opposite values of κ . If the (114) formula was not the same for opposite values of κ , we should not be able to get four polynomial radial functions with only one condition which gives the quantification of the energy levels. The Standard Model has a precise answer using vacuum polarization. But the calculation must be revised, both to avoid divergences and to employ the improved wave equation which accounts also for weak interactions [33] [34]. Since the Lamb shift is of the same order as the hyperfine structure coming from the interaction between the magnetic moment of the proton and that of the electron, a true calculation must account for the origin of the magnetic moment of the proton from the waves of the three quarks inside, and for the true potential seen by the electron wave, with both the electric charges and the potential vector of the moving charges.